1. Lecture 26 Prof. Dr. M. Junaid Mughal Mathematical Statistics 1

2. Last Class Normal Distribution Approximation to Binomial Distribution 2

3. Today’s Agenda Approximation to Binomial Distribution Gamma Distribution Exponential Distribution 3

4. Binomial Distribution 4 Suppose out of a sample of 1,500 person we want to assess whether the representation of foreigners in the sample is accurate. We know that about 12% of population is foreigners. Let X be the number of foreigners in the sample, what is the probability that the sample contains 170 or fewer foreigners?

5. Binomial Distribution 5 It turns out that as n gets larger, the Binomial distribution looks increasingly like the Normal distribution. Consider the following Binomial histograms, Binomial distribution with p = 0.1

6. Binomial Distribution 6

7. 7

8. Parameters of the Approximating Distribution 8 The approximating Normal distribution has the same mean and standard deviation as the underlying Binomial distribution. Thus, if X ~ B(n; p), having mean E[X] = np and standard deviation SD(X) = √np(1 - p) = sqrt(npq), It can be approximated by Normal distribution with

9. When is the approximation appropriate? 9 The farther p is from 0.5, the larger n needs to be for the approximation to work. Thus, as a rule of thumb, only use the approximation if

10. Behavior of the Approximation as a Function of p, for n = 100 10

11. Calculations with the Normal Approximation 11 Recall the problem we set out to solve: – P(X <170); where X ~B(1500; 0.12) How do we calculate this using the Normal approximation? If we were to draw a histogram of the B(1500; 0.12) distribution with bins of width one, P(X < 170) would be represented by the total area of the bins spanning

12. Calculations with the Normal Approximation 12 Thus, using the approximating Normal distribution Y ~ N(170; 0.12), we calculate P(X ≤ 170) ≈ P(Y <170.5) = 0.2253 For reference, the exact Binomial probability is 0.2265, so the approximation is apparently pretty good

13. Calculations with the Normal Approximation 13 The addition of 0.5 in the previous slide is an example of the continuity correction which is intended to refine the approximation by accounting for the fact that the Binomial distribution is discrete while the Normal distribution is continuous. In general, we make the following adjustments:

14. Normal Approximation to the Binomial 14

15. Comparison 15

16. Comparison 16

17. Example The probability that a patient recovers from a rare blood disease is 0.4. If 100 people are known to have contracted this disease, what is the probability that less than 30 survive? 17

18. Example A multiple-choice quiz has 200 questions each with 4 possible answers of which only 1 is the correct answer. What is the probability that sheer guesswork yields from 25 to 30 correct answers for 80 of the 200 problems about which the student has no knowledge? 18

19. Gamma Distribution 19

20. Properties of Gamma Function 20

21. Gamma Distribution 21

22. Gamma Distribution 22

23. Properties of Gamma Function 23 Proof:

24. Mean and Variance 24 The mean and variance of Gamma distribution are PTO

25. Mean and Variance 25

26. Exponential Distribution 26

27. Mean and Variance 27 The mean and variance of Exponential distribution are

28. References Probability and Statistics for Engineers and Scientists by Walpole Schaum outline series in Probability and Statistics Summary Gamma Distribution Exponential Distribution 28